Transcription of Affine Transformations - Clemson University
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Affine Transformations - Clemson University
A P P E N D I XCAffine The need for geometric Affine matrix representation of the linear Homogeneous 3D form of the Affine THE NEED FOR GEOMETRIC TRANSFORMATIONSOne could imagine a computer graphics system that requires the user to construct ev-erything directly into a single scene. But, one can also immediately see that this wouldbe an extremely limiting approach. In the real world, things come from various placesand are arranged together to create a scene. Further, many of these things are themselvescollections of smaller parts that are assembled together. We may wish to define one objectrelative to another for example we may want to place a hand at the end of an arm. Also,it is often the case that parts of an object are similar, like the tires on a car. And, eventhings that are built on scene, like a house for example, are designed elsewhere, at a scalethat is usually many times smaller than the house as it is built.
Because ma-trix multiplication is associative, we can remove the parentheses and multiply the three matrices together, giving a new matrix M = RHS. Now we can rewrite our transform x0= (RHS)x = Mx If we have to transform thousands of points on a complex model, it is clearly easier to
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